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Chapter 9: Problem 26

Reduce each of the following fractions as completely as possible. $$\frac{-3 a}{6 a-18}$$

Short Answer

Expert verified
\( \frac{-a}{2(a - 3)} \)

Step by step solution

01

Identify the common factor in the denominator

First, factor the expression in the denominator. Notice that both terms in the denominator have a common factor of 6. The denominator can be rewritten as: \[6a - 18 = 6(a - 3) \]
02

Simplify the fraction

Now that we have factored the denominator, rewrite the original fraction by substituting the factored form: \[ \frac{-3a}{6(a - 3)} \]
03

Reduce the fraction

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor. The greatest common divisor of \(-3a\) and \(6a\) is 3a. So reduce the fraction: \[ \frac{-3a}{6(a - 3)} = \frac{-3a}{6(a - 3)} = \frac{-a}{2(a - 3)} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Factor
Understanding common factors is crucial when simplifying fractions. A common factor is a number or expression that divides two or more numbers or expressions without leaving a remainder. In our exercise, we first identified common factors in the denominator, which were 6 and (a - 3). Looking for these shared factors helps to simplify the fraction easily.

For instance, in the expression 6a - 18, both 6 and 18 share a common factor of 6. That means 6 can divide both numbers completely. By factoring out this common factor, we can make the expression simpler, which brings us to our next concept.
Factoring Expressions
Factoring expressions involves breaking down a complex expression into simpler factors that, when multiplied, give back the original expression. This is an essential step in reducing fractions.

In the exercise, we factored the denominator 6a - 18 into 6(a - 3). This was achieved by identifying the common factor of 6 and then rewriting the expression accordingly. Once the denominator is factored, it becomes easier to see how the numerator and denominator relate, which will help in reducing the fraction.

Factoring greatly simplifies complex algebraic equations and is an important skill for solving many types of math problems.
Greatest Common Divisor
The greatest common divisor (GCD) is the largest number or expression that can be evenly divided into both the numerator and the denominator of a fraction.

In the given exercise, the GCD of \(-3a\) and 6(a - 3) is 3a. Dividing both the numerator and the denominator by this GCD allows us to reduce the fraction to its simplest form, \(-a/2(a-3)\).

Finding the GCD requires examining both the numerical and variable components of the terms involved. This step ensures that the fraction is as simple as possible and easy to understand when solving problems or further manipulating the expression.

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Most popular questions from this chapter

In Exercises \(1-62,\) perform the indicated operations and simplify your answer as completely as possible. $$2-\frac{x}{x-1}$$

Perform the indicated operations and simplify as completely as possible. $$\left(\frac{x}{3} \cdot \frac{x}{4}\right) \cdot \frac{12}{x^{2}-3 x}$$

In Exercises \(1-62,\) perform the indicated operations and simplify your answer as completely as possible. $$\frac{\frac{2}{a}-\frac{1}{a+3}}{\frac{a+6}{a}}$$

Solve each of the problems algebraically. Be sure to label what the variable represents. In a certain voting district the ratio of Republicans to Democrats is 4 to \(7 .\) If there are 2520 Democrats, how many Republicans are there?

Perform the indicated operations and simplify as completely as possible. $$\frac{x}{2} \div \frac{2}{x} \cdot \frac{x^{2}-16}{x^{2}-4 x}$$
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